Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2047 |
256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(5,7)(8,11,12,24,13,25,28,39)(9,15,16,32,17,22,26,37)(10,19,20,36,21,23,27,38)(14,29,18,34,30,31,33,35);(8,10,13,21)(9,14,17,30)(11,23,25,19)(12,27,28,20)(15,31,22,29)(16,33,26,18)(24,36,39,38)(32,34,37,35);(2,3)(4,5)(6,7)(8,9,13,17)(10,18,21,33)(11,22,25,15)(12,26,28,16)(14,27,30,20)(19,35,23,34)(24,32,39,37)(29,36,31,38)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^2,c^b=cf,d^b=dg,e^c=eg,f^c=fg,h^c=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2047, C2×#70340, #10822990, … |
| maximal normal subgroup of | D3×#2047, C2×#70340, #10822990, … |
| maximal quotient of | #10698794, #10698796, #10698801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2916);